Bayesian Change-Point Analysis for Atomic Force Microscopy and Soft Material Indentation

Abstract

Summary. Material indentation studies, in which a probe is brought into controlled physical contact with an experimental sample, have long been a primary means by which scientists characterize the mechanical properties of materials. More recently, the advent of atomic force microscopy, which operates on the same fundamental principle, has in turn revolutionized the nanoscale analysis of soft biomaterials such as cells and tissues. The paper addresses the inferential problems that are associated with material indentation and atomic force microscopy, through a framework for the change‐point analysis of pre‐contact and post‐contact data that is applicable to experiments across a variety of physical scales. A hierarchical Bayesian model is proposed to account for experimentally observed change‐point smoothness constraints and measurement error variability, with efficient Monte Carlo methods developed and employed to realize inference via posterior sampling for parameters such as Young's modulus, which is a key quantifier of material stiffness. These results are the first to provide the materials science community with rigorous inference procedures and quantification of uncertainty, via optimized and fully automated high throughput algorithms, implemented as the publicly available software package BayesCP. To demonstrate the consistent accuracy and wide applicability of this approach, results are shown for a variety of data sets from both macromaterials and micromaterials experiments—including silicone, neurons and red blood cells—conducted by the authors and others.

Design and fabrication of a flexible MEMS-based electro-mechanical sensor array for breast cancer diagnosis

“…For n number of data sets in calibration, output resistance R s , and contact F have a linear relationship to the sensor deformation δ s , when contact occurs at the k th index. Thus: $${\left(R_s\right)}_i=\{\begin{array}{cc}{\alpha }_{11}+{\alpha }_{12}{\left({\delta }_s\right)}_i+{\epsilon }_1\hfill & ifi\le k\hfill \\ {\alpha }_{21}+{\alpha }_{22}{\left({\delta }_s\right)}_i+{\epsilon }_2\hfill & ifk+1\le i\le n\hfill \end{array}\phantom{true\}}$$$${\mathrm{normalF}}_i=k_s{\delta }_s=\{\begin{array}{cc}{k}_{s}true[\frac{{\left(R_s\right)}_i-{\alpha }_{11}+{\epsilon }_1}{{\alpha }_{12}}true]\hfill & ifi\le k\hfill \\ k_{s}true[\frac{{\left(R_s\right)}_i-{\alpha }_{21}+{\epsilon }_2}{{\alpha }_{22}}true]\hfill & ifk+1\le i\le n\hfill \end{array}\phantom{true\}}$$ where the values α j = [α j1 (Ω) α j2 (Ω/μm)] ; (j=1,2) are linearly regressed parameters in the non-contact and contact regime, respectively, k s is the calibrated spring constant of the sensor and ε j ; (j=1,2) are the errors that typically have different distribution depending on the material interaction with the sensor [51]. In this model, ε 1 is caused by the viscous interaction between the sensor and PBS solution covering the tissue, while ε 2 are primarily results from sensor-tissue friction.…”

Determination of Mechanical Properties of Spatially Heterogeneous Breast Tissue Specimens Using Contact Mode Atomic Force Microscopy (AFM)

“…, 2 a probabilistic extension of which has been recently explored by Rudoy et al. 23 This approach relies on the assumption that the AFM force curve can be successfully segmented into a non-contact and contact region, governed by a two-regime regression model, as stated below: $$F_i=\{\begin{array}{cc}{\beta }_{11}+{\delta }_i{\beta }_{12}+{ϵ}_1\hfill & \text{if}i\le k\left(\text{non-contact}\right)\hfill \\ {\beta }_{21}+f_i{\beta }_{22}+{ϵ}_2\hfill & \text{if}k+1\le i\le n\left(\text{contact}\right)\hfill \end{array}\phantom{true\}}$$ where F i is the observed force obtained by multiplying the deflection d i by the probe spring constant k c , δ i is the probe deflection d subtracted from z-position z , i.e. δ i = z i – d i .…”

Probabilistic Estimation of Mechanical Properties of Biomaterials Using Atomic Force Microscopy

“…7. In general, ${\sigma }_1^2\ne {\sigma }_2^2$, since ${\sigma }_1^2$ results from the viscous interactions between the probe and the PBS solution, while ${\sigma }_2^2$ depends primarily on the probe-tissue frictional forces [22]. ${\beta }_j={\left[\begin{array}{cc}\hfill {\beta }_{j1}\hfill & \hfill {\beta }_{j2}\hfill \end{array}\right]}^T$ ; j = 1, 2 are the regression coefficients in the non-contact and contact regime respectively.…”

“…14 are multiples of the probe compliance s . When s is observed without errors, the estimation approach proposed in this work reduces to the regular Bayesian Changepoint model [22],[36], since ξ 1 , …, ξ n are fixed and observed. However, variations in the probe calibration results as evidenced by the results in Table I indicate that incorporation of some notion of randomness on the nature of s is necessary.…”

“…Rudoy et al [22] have recently proposed rigorous statistical methods using Bayesian analysis to obtain estimates of the elastic modulus due to the contact-point uncertainty. While these methods successfully extract the material property in soft samples after accounting for uncertainty due to the contact point, such methods do not take into consideration probe spring constant variations.…”